Current Organic Chemistry - Volume 19, Issue 4, 2015

The present status of what has been known and understood about the concept of cross-conjugation is reviewed first and it has been shown how the structure-activity relationship in acyclic and monocyclic conjugated systems can be understood without elaborated formalism but by gathering the knowledge of the basic HMO of linear polyenes. Then it is shown that the mode of electron flow caused by hetero-atom substitution to these hydrocarbons as depicted by the curly arrows on the structural formula is correlated well with the topological structure of cross-conjugated system. Mathematical relations between the conventional HMO and resonance theories especially related to the Kekule structure of conjugated hydrocarbons are explained. The least amount of graph-theoretical techniques, such as the topological index, Z, is also explained. It will be understood how the global trend of the direction and extent of the π-electron flow in acyclic conjugated hydrocarbon networks can be predicted by the diagrammatic method of organic electron theory and how cross-conjugation is an important concept in organic chemistry. Finally several new research trends and observations are introduced and discussed from other than organic chemistry, such as “quantum interference” in physics.

A graph property P(G), written as a sequence of numbers counting its local partitions, can be expressed as a counting polynomial P(G,x) with the exponents of the indeterminate x showing the extent of partitions p(G), υ p(G) = P(G) while the coefficients are related to the number of partitions occurrence. Definitions and properties of Cluj CJ(G, x) and Ω(G, x) polynomials and their relation with other topological descriptors are presented. Analytical relations for calculating these polynomials and corresponding topological indices in some classes of PAHs are derived. The ability of topological descriptors to predict various physico-chemical and biological properties is reviewed. Omega polynomial applied to small fullerenes provided a single number descriptor useful in predicting their energetic stability.

Polycyclic aromatic compounds ( PAC ) are a huge class of organic molecules that can be found everywhere. They called attention of scientists by their interesting and sometimes peculiar physical and chemical properties. That is why, there are a number of research articles and books in which their synthesis, properties, reactions, and theoretical investigations have been communicated. In this survey will be outlined several simple methods for assessing the partition of π-electrons in rings of some classes of PAC.

The cut method is a powerful tool for the investigation of distance-based (and some other) molecular structure-descriptors. In this paper a survey on the recent developments of the method is given. The instances of the standard cut method for the Wiener index, the Szeged index, the PI index, the generalized terminal Wiener index, the Gutman index, the edge-Wiener index, and the edge-Szeged index are described, where a standard cut method is a method that applies to partial cubes. It is pointed out that the standard cut method was recently independently discovered a couple of times. Numerous proper extensions of the standard cut method are presented. The method extends to l1-graphs, graphs with a non-trivial canonical metric representation, graphs with transitive relation Θ, and partial Hamming graphs. The instances of these extended cut methods include the Wiener index, the degree distance, distance moments, and the colored Wiener index.

With the rise of the nanoscience era in general and that of graphene structure and nanopatterns especially their precursors by organic compounds and specifically the benzenoid systems become most regarded due to the benchmarking information they provide and transfer from nano-to-macro chemistry and materials. In this context, chemical graph theory and methodology are reviewed from the fundaments, via a historical development of a molecular graph and of their motivation, to the algebraic polynomial formulation to the powerful quantum representation in the close relation with the celebrated Hückel molecular orbital method; in all these stages molecular graph theory benefits from the adjacency information of atoms in molecules, in terms of vertices linked by edges representing the chemical bonds. The chemical topology to chemical reactivity passage is here advanced with the aim of coloring of matrix of atoms’ adjacencies in molecules by standard reactivity indices as of electronegativity and hardness, along with the recently introduced chemical power and the electrophilicity, all being able to clarify bonding in a mechanistic/causal manner as based on the first principles of quantum chemistry. The actual topo-reactivity coloring methodology further uses the fragments of atoms in molecules as centered on the chemical bonding under concern (the so called specific-bond-in-adjacency, SBA), either single or double and when the hetero-distinctions are made possible. As an application, a series of paradigmatic polycyclic aromatic hydrocarbons (PAHs) is analyzed from the present topo-reactivity methodology both for in gas and liquid phase to yield the conclusion that the electrophilicity is the actual driving force triggering their reactivity, thereby refining previous molecular-orbital electronegativity-based approaches. The actual review and topo-reactivity methods are of general value to be used in the XXIth-century quantum nanochemistry research and developments in organic chemistry and chemistry of materials.

The distinction between interdisciplinarity, multidisciplinarity, and transdisciplinarity is emphasized. Chemistry evolved later as a science than physics, so that in the early 19th century renowned philosophers denied the possibility of mathematics being useful for chemical problems. Later, with the advent of quantum mechanics, chemistry was viewed as reducible to physics. It is argued, however, that mathematical chemistry – discrete mathematics and graph theory – can explain chemical phenomena and concepts without the intermediacy of physics. Isomer enumeration of cyclic and acyclic organic compounds (as molecular graphs) contributed to the birth of graph theory. Aromatic heterocyclic compounds can be described and enumerated according to three types of atoms forming the aromatic ring, according to the number (0, 1, or 2) of π-electrons in the non-hybridized orbital. Benzenoids and diamondoids can be classified, described and enumerated with the help of dualists. In polycyclic benzenoids, various π-electron partitions are possible, and among them the Clar-sextet partition is the most unequal. Topological indices that associate numbers with molecular structures are instruments for quantitative structure-property or structure-activity relationships (QSPR and QSAR, respectively) and drug design. Reaction graphs and synthon graphs are also mentioned. Reciprocal interaction – from chemistry to mathematics – accounts for the discovery of new (3,g)-cages and upper/lower bounds of various energy graphs or graph connectivities.